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Properties

Label 99.4.f.c
Level $99$
Weight $4$
Character orbit 99.f
Analytic conductor $5.841$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,4,Mod(37,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.f (of order \(5\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.29283765625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{2} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \cdots + 3) q^{4}+ \cdots + (26 \beta_{6} - 3 \beta_{5} + 26 \beta_{4} + \cdots + 7) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_{2} - \beta_1 + 1) q^{2} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \cdots + 3) q^{4}+ \cdots + (242 \beta_{7} - 1071 \beta_{6} + \cdots - 843) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 7 q^{2} + 3 q^{4} + 7 q^{5} - 35 q^{7} - 47 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 7 q^{2} + 3 q^{4} + 7 q^{5} - 35 q^{7} - 47 q^{8} + 40 q^{10} - 67 q^{11} - 65 q^{13} + 196 q^{14} - 377 q^{16} + 31 q^{17} + 148 q^{19} - 342 q^{20} + 647 q^{22} + 12 q^{23} - 201 q^{25} + 140 q^{26} - 42 q^{28} + 199 q^{29} - 361 q^{31} - 324 q^{32} - 298 q^{34} - 237 q^{35} + 81 q^{37} + 52 q^{38} + 532 q^{40} + 31 q^{41} - 650 q^{43} + 1208 q^{44} + 1204 q^{46} - 857 q^{47} + 1375 q^{49} + 147 q^{50} - 590 q^{52} + 1493 q^{53} - 1583 q^{55} + 1560 q^{56} + 1392 q^{58} - 676 q^{59} - 525 q^{61} - 2456 q^{62} + 471 q^{64} - 1790 q^{65} + 86 q^{67} - 710 q^{68} - 144 q^{70} - 1143 q^{71} - 2155 q^{73} + 1476 q^{74} - 242 q^{76} + 2015 q^{77} - 861 q^{79} + 1916 q^{80} - 3497 q^{82} - 52 q^{83} + 2383 q^{85} - 1061 q^{86} + 4543 q^{88} - 3782 q^{89} + 135 q^{91} + 2450 q^{92} + 702 q^{94} + 1317 q^{95} - 1344 q^{97} - 2740 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 10x^{6} - 19x^{5} + 109x^{4} + 171x^{3} + 810x^{2} + 729x + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 280\nu ) / 981 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 171 ) / 109 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 1261\nu^{2} ) / 8829 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 10\nu^{6} + 19\nu^{5} - 109\nu^{4} + 1090\nu^{3} - 810\nu^{2} - 729\nu - 6561 ) / 8829 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -19\nu^{7} + 19\nu^{6} - 190\nu^{5} + 1090\nu^{4} - 2071\nu^{3} - 3249\nu^{2} - 15390\nu - 13851 ) / 79461 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\nu^{7} + 3781\nu^{2} ) / 8829 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + 10\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 9\beta_{6} + 10\beta_{5} + 9\beta_{4} + 9\beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 19\beta_{7} + 90\beta_{6} + 19\beta_{5} - 19\beta_{4} + 19\beta_{3} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 109\beta_{3} - 171 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 981\beta_{2} - 280\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1261\beta_{7} - 3781\beta_{4} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
2.86504 + 2.08157i
−2.05602 − 1.49379i
0.785330 − 2.41700i
−1.09435 + 3.36805i
0.785330 + 2.41700i
−1.09435 − 3.36805i
2.86504 − 2.08157i
−2.05602 + 1.49379i
−0.747004 − 0.542730i 0 −2.20868 − 6.79761i 7.05908 − 5.12872i 0 0.239524 + 0.737179i −4.32202 + 13.3018i 0 −8.05666
37.2 4.17405 + 3.03263i 0 5.75376 + 17.7083i −6.98613 + 5.07572i 0 0.513765 + 1.58121i −16.9313 + 52.1091i 0 −44.5532
64.1 −0.903364 + 2.78027i 0 −0.441690 − 0.320907i −1.90871 − 5.87440i 0 −25.9914 − 18.8838i −17.6291 + 12.8083i 0 18.0567
64.2 0.976313 − 3.00478i 0 −1.60339 − 1.16493i 5.33576 + 16.4218i 0 7.73807 + 5.62204i 15.3824 − 11.1760i 0 54.5532
82.1 −0.903364 − 2.78027i 0 −0.441690 + 0.320907i −1.90871 + 5.87440i 0 −25.9914 + 18.8838i −17.6291 − 12.8083i 0 18.0567
82.2 0.976313 + 3.00478i 0 −1.60339 + 1.16493i 5.33576 − 16.4218i 0 7.73807 − 5.62204i 15.3824 + 11.1760i 0 54.5532
91.1 −0.747004 + 0.542730i 0 −2.20868 + 6.79761i 7.05908 + 5.12872i 0 0.239524 − 0.737179i −4.32202 − 13.3018i 0 −8.05666
91.2 4.17405 − 3.03263i 0 5.75376 − 17.7083i −6.98613 − 5.07572i 0 0.513765 − 1.58121i −16.9313 − 52.1091i 0 −44.5532
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.4.f.c 8
3.b odd 2 1 11.4.c.a ✓ 8
11.c even 5 1 inner 99.4.f.c 8
11.c even 5 1 1089.4.a.y 4
11.d odd 10 1 1089.4.a.bh 4
12.b even 2 1 176.4.m.c 8
33.d even 2 1 121.4.c.i 8
33.f even 10 1 121.4.a.f 4
33.f even 10 2 121.4.c.b 8
33.f even 10 1 121.4.c.i 8
33.h odd 10 1 11.4.c.a ✓ 8
33.h odd 10 1 121.4.a.g 4
33.h odd 10 2 121.4.c.h 8
132.n odd 10 1 1936.4.a.bl 4
132.o even 10 1 176.4.m.c 8
132.o even 10 1 1936.4.a.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.c.a ✓ 8 3.b odd 2 1
11.4.c.a ✓ 8 33.h odd 10 1
99.4.f.c 8 1.a even 1 1 trivial
99.4.f.c 8 11.c even 5 1 inner
121.4.a.f 4 33.f even 10 1
121.4.a.g 4 33.h odd 10 1
121.4.c.b 8 33.f even 10 2
121.4.c.h 8 33.h odd 10 2
121.4.c.i 8 33.d even 2 1
121.4.c.i 8 33.f even 10 1
176.4.m.c 8 12.b even 2 1
176.4.m.c 8 132.o even 10 1
1089.4.a.y 4 11.c even 5 1
1089.4.a.bh 4 11.d odd 10 1
1936.4.a.bk 4 132.o even 10 1
1936.4.a.bl 4 132.n odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 7T_{2}^{7} + 31T_{2}^{6} - 71T_{2}^{5} + 319T_{2}^{4} - 78T_{2}^{3} + 1664T_{2}^{2} + 2816T_{2} + 1936 \) acting on \(S_{4}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 7 T^{7} + \cdots + 64577296 \) Copy content Toggle raw display
$7$ \( T^{8} + 35 T^{7} + \cdots + 156816 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 3138428376721 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 2905210000 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 9608691844521 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 90104309844241 \) Copy content Toggle raw display
$23$ \( (T^{4} - 6 T^{3} + \cdots + 49883584)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 97\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{4} + 325 T^{3} + \cdots - 1288748736)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 70\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 66\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{4} - 43 T^{3} + \cdots - 8869996224)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 32\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1891 T^{3} + \cdots - 2046678844)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
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